A note on estimation of population mean in sample survey using auxiliary information

Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.9, 2013-Special issue, International Conference on Recent Trends in Applied Sciences with Engineering Applications
9
A Note on Estimation of Population Mean in Sample Survey using
Auxiliary Information
R. K. Gangele and S.K. Choubey
Department of Mathematics and Statistics
Dr. H.S. Gour Central University, Sagar – 470003 (M.P.) INDIA
E-mail- rkgangele@gmail.com
Abstract
In this paper we have suggested a class of estimators for population mean using auxiliary information in two-
phase sampling. When the population mean X is not known, a class of estimators for finite population mean
∑=
=
N
1i
iy
N
1
Y of the study variable y has been suggested. Expressions of bias and mean squared error are
obtained upto the first order of approximation. Asymptotically optimum estimators (AOE’s) are also identified
with its mean squared error formula. We found that proposed class of estimator are better than usual ratio and
other estimators.
1.1 Introduction
Whenever there is auxiliary information available, the investigator wants to use it in the method of
estimation which yields larger efficiency. Ratio, regression and product methods of estimation are good
examples in this context. When the population mean of the auxiliary variable is known a large number of
estimators for the population mean of the study variable is available in the literature for instance see Singh,
H. P. (1986) and Singh, S. (2003) and the references cited therein. It is observed in the literature that the
efficiency of the ratio and product estimators may be increased to the efficiency of regression estimator by
making use of prior knowledge of = , where is the correlation coefficient between the study variable
and auxiliary variable , and are the coefficients of variation of and respectively. Sacrificing the
consistency of estimators researchers including Upadhyaya and Singh (1985), Srivastava (1974), Prasad (1989)
and Lui (1990) have found the way by which the efficiency an estimator can be increased beyond the regression
estimator of the mean in case of known population mean .
When the population mean of the auxiliary variable is not known, it is often estimated from a
preliminary large sample on which only the auxiliary variable is observed. The value of the population mean
of the auxiliary variable is then replaced by this estimate. This procedure is known as the double sampling or
two-phase sampling. Throughout, samples have been drawn by the method of simple random sampling without
replacement (SRSWOR). The sample survey statisticians have presented several modifications of the classical
two-phase sampling ratio and product estimators and studied their properties. They have shown that the
efficiency of their estimators can be increased maximum upto usual two-phase sampling regression estimator by
using of the well known optimum choice = , for instance see Srivastava (1970, 81), Gupta (1978) and
Adhvaryu and Gupta (1983).
In this paper we have made an effort to suggest a class of estimators using auxiliary information in two-
phase sampling and studied its properties. When the population mean is not known, a class of estimators for
finite population mean ∑=
=
N
1i
iy
N
1
Y of the study variable y has been suggested. A large number of estimators
are identified as the member of the proposed class of estimators. Expressions of bias and mean squared error are
obtained upto the first order of approximation. Asymptotically optimum estimators (AOE’s) are also identified
with its mean squared error formula.
1.2 Two-Phase Sampling Procedure
Consider the finite population = , , … , of identifiable units. Let and be the variable
under study and the auxiliary variable respectively. Further let be the unknown real variable value of and
be the known variable value of associated with , = 1,2, … , . The problem of estimating the population
mean ∑=
=
N
1i
iy
N
1
Y of the study variable when the population mean ∑=
=
N
1i
ix
N
1
X of the auxiliary variable

10
is known has been dealt at a grater length. However, in many practical situations when the population mean
is unknown a prior, the procedure of double sampling is used. Allowing, simple random sampling without
replacement (SRSWOR) design in each phase, the two-phase (or double) sampling scheme will be as follows:
(i) The first-phase sample ⊂ of fixed size ! is drawn to observe only in order to furnish a
good estimate of the population mean .
(ii) Given , the second phase sample ⊂ of fixed size ! is drawn to observe only.
Let # =
$%
∑$%
' , =
$
∑$
' and # =
$
∑$
' .
Further we write
= 1 + )*
# = 1 + )
# = 1 + ) 1.2.1
Such that
2
1
21
2
1
20
2
10
2
1
2
2
22
1
22
0
210
11
)(,
11
)(
11
)(,
11
)(
,
11
)(,
11
)(
0)()()(
xx
xx
xy
C
Nn
eeECk
Nn
eeE
Ck
Nn
eeEC
Nn
eE
C
Nn
eEC
Nn
eE
and
eEeEeE






−=





−=






−=





−=






−=





−=
===
where
=
+
,
, =
+
-
, . =
/
∑ −'
and
. =
/
∑ −' .
1.3 Proposed Class of Estimators
Motivated by Gangele (1995) and Gupta (1978), we define a class of estimators for population mean
in two-phase sampling as
1
23 = ∑ 45
'*
#
#%
6
, 1.3.1
where 4′
= 0,1,2,3 are suitably chosen constants whose sum need not be unity, 9 is a suitably chosen scalar
takes value +1 for product-type estimator and −1 for ratio-type estimator; # is a first phase sample mean based
on ! observations and , # are second phase sample means of , respectively based on ! observations.
Some members of the proposed class of estimators are shown in Table-1.3.1.
1.2.2

11
Table-1.3.1
Some members of the class of estimators 1
23
S.
No.
Estimator Choice of constants
4* 4 4 45 9
1 The mean per unit
=
$
∑$
'
1 0 0 0 -
2 The classical two-phase sampling
ratio estimator
:3 =
#%
#
0 1 0 0 -1
3 The classical two-phase sampling
product estimator
;3 =
#
#%
0 1 0 0 1
4 Srivastava (1970) estimator
3 =
#%
#
6 0 1 0 0 −9
5 Chakraberty (1968), Vos (1980),
Adhvaryu and Gupta (1983) type-
estimator
3 = 4* + 1 − 4* <
#
#
=
4* 1 − 4* 0 0 -1
6 Vos (1980), Adhvaryu and Gupta
(1983) type estimator
53 = 4* + 1 − 4* <
#
#
=
4* 1 − 4* 0 0 1
4, 9 being constants.
1.4 Bias and Mean Squared Error
To obtain the bias and mean squared error of the proposed class of estimators 1
23, we express it in
terms of )′ we have
1
23 = ∑ 45
'* 1 + )* 1 + ) 6
1 + )′ / 6
.
or
1
23 = 1 + )* ∑ 45
'* 1 + ) 6
1 + )′ / 6
. 1.4.1
We assume that the sample size ! and ! ! > ! are large enough so that
|) | < 1 and A)′
A < 1.
i.e. B
#/-
-
B < 1 and B
#%/-
-
B < 1.
and 1 + ) and 1 + )′
are expandable.
Expanding the right hand side of (1.4.1) we have
1
23 = ∑ 45
'* 1 + )* C1 + 9) +
6 6/
) + ⋯ E ×
C1 − 9)′
+
6 6G
)′
+ ⋯ E
or
1
23 = H 4
5
'*
I1 + )* + 9) + 9)*) +
9 9 − 1
2
) − 9)′
− 9)*)′
− 9 ) )′
+
9 9 + 1
2
)′
+
9 9 − 1
2
)*) − 9 )*) )′
−
9 9 − 1
2
) )′
+
9 9 + 1
2
)*)′
+
9 9 + 1
2
) )′
+ ⋯ J.
Neglecting terms of )′ having power greater than two we have
1
23 = ∑ 45
'* K1 + )* + 9 ) − )′
+ 9 )*) − )*)′
− 9 ) )′
+
6 6/
) +
9 9+12)1′2.or
1
23 − = K∑ 45
'* K1 + )* + 9 ) − )′
+ 9 )*) − )*)′
− 9 ) )′
+
6 6/
) +
9 9+12)1′2−1.

12
1.4.2
Taking expectation of both sides of (1.4.2) and using the expected values given by (1.2.2) we get the
bias of 1
23 to the first degree of approximation as
L 1
23 = M∑ 45
'* M
1 +
$
−
$′ 9 +
6 6/
$
−
−
6 6/
$′ −
N − 1N
= K∑ 45
'* K1 +
$
−
$′ 9 +
$
−
$′
6 6G
O − 1O
i.e.
L 1
23 = K∑ 45
'* K1 +
$
−
$′ 9 +
6/
O − 1O. 1.4.3
Squaring both sides of (1.4.2) and neglecting term of )′ having power greater than two we have
PQR
ST − PQ U
= PQU
V
W
W
W
W
W
W
W
W
W
W
W
W
W
W
X
Y + ∑ SZ
U[
Z'
]
^
_
^
`
Y + Ua + UZbcaY − aY
′
d
+eZbcaaY − aaY
′
d + a
U
+ZU
bU
caY − aY
′
d
U
− UZU
bU
aYaY
′
+Zb Zb − Y aY
U
+ Zb Zb + Y aY
′U
f
^
g
^
h
+U ∑ SZ
[
Z Zij ' Sj
]
^
_
^
`
Y + Ua + Z + j bcaY − aY
′
d
+a
U
+ U Z + j caaY − aaY
′
d
− Z − j U
bU
aYaY
′
+
b ZGj
U
b Z + j − Y aY
U
+
b ZGj
U
b Z + j − Y aY
′U
f
^
g
^
h
−U ∑ k
Y + a + ZbcaY − aY
′
d
+ZbcaaY − aaY
′
d
−ZU
bU
aYaY
′
+
Zb Zb/Y
U
aY
U
+
Zb ZbGY
U
aY
′U
l[
Z'
m
n
n
n
n
n
n
n
n
n
n
n
n
n
n
o
.
1.4.4
Taking expectation of both sides of (1.4.4) using the result in (1.2.2) we get the MSE of 1
23 to the first degree of
approximation as
p.q 1
23 = r1 + ∑ 45
'* s + 2 ∑ 45
it '* 4ts t − 2 ∑ 4 L5
'* u, 1.4.5
where
s = K1 +
$
− +
$
−
$′ 9 4 + 2 9 − 1 O = 0,1,2,3 ,
s t = K1 +
$
− +
$
−
$′
6 Gt
4 + 9 + w − 1 O
< w = 0,1,2,3 ,
L = K1 +
$
−
$′ 9 +
6/
O = 0,1,2,3 .
Expression (1.4.5) can be rewritten as
p.q 1
23 = M
1 + 4*s* + 4 s + 4 s + 45s5
+2 4*4 s* + 4*4 s* + 4*45s*5 + 4 4 s + 4 45s 5 + 4 45s 5
−2 4*L* + 4 L + 4 L + 45L5
N,
1.4.6
where
L* = 1,
L = K1 +
$
−
$′ 4 C +
6/
E O,
L = K1 +
$
−
$′ 24 C +
6/
E O,
L5 = K1 +
$
−
$′ 34 C +
56/
E O,
s* = K1 +
$
− O,
s = K1 +
$
− +
$
−
$′ 9 4 + 29 − 1 O,
s = K1 +
$
− +
$
−
$′ 29 4 + 49 − 1 O,
(

13
s5 = K1 +
$
− +
$
−
$′ 39 4 + 69 − 1 O,
s* = K1 +
$
− +
$
−
$′
6
4 + 9 − 1 O,
s* = K1 +
$
− +
$
−
$′ 9 4 + 29 − 1 O,
s*5 = K1 +
$
− +
$
−
$′
56
4 + 39 − 1 O,
s = K1 +
$
− +
$
−
$′
56
4 + 39 − 1 O,
s 5 = K1 +
$
− +
$
−
$′ 29 4 + 49 − 1 O,
and
s 5 = K1 +
$
− +
$
−
$′
y6
4 + 59 − 1 O.
Differentiating the p.q 1
23 given in (1.4.6) with respect to 4′
, = 0,1,2,3 partially and equating them to
zero we have












=
























3
2
1
3
2
1
0
3231303
2321202
.1312101
0302010 1
B
B
B
AAAA
AAAA
AAAA
AAAA
α
α
α
α
1.4.7
Solving (1.4.6) we get the optimum values of constants 4′
, = 0,1,2,3 as
∆
∆
=
∆
∆
=
∆
∆
=
∆
∆
=
∗∗
∗∗
3
3
2
2
1
1
0
0 ,
αα
αα
1.4.8
where












=∆
3231303
2321202
.1312101
0302010
AAAA
AAAA
AAAA
AAAA












=∆
323133
232122
.131211
030201
0
1
AAAB
AAAB
AAAB
AAA












=∆
323303
232202
.1312101
03020
1
1
AABA
AABA
AABA
AAA












=∆
331303
2321202
.131101
03010
2
1
ABAA
ABAA
ABAA
AAA
.
1
3231303
221202
.112101
02010
3












=∆
BAAA
BAAA
BAAA
AAA
Putting the optimum values 4∗
of 4 = 0,1,2,3 in (1.6) we get the minimum MSE of proposed estimator 1
23
as
p.q 1
23 = |1 − 4*
∗
− 4∗
L − 4∗
L − 45
∗
L5}. 1.4.9
or
p.q 1
23 = K1 −
∆•G∆%€%G∆•€•G∆‚€‚
∆
O. 1.4.10
Thus we established the following theorem.
Theorem 1.4.1. To the first degree of approximation,
p.q 1
23 ≥ K1 −
∆•G∆%€%G∆•€•G∆‚€‚
∆
O,

14
with equality holding if
4*
∗
=
∆•
∆
, 4∗
=
∆%
∆
, 4∗
=
∆•
∆
, 45
∗
=
∆‚
∆
.
It is to be noted that the biases and mean squared errors of the estimators belonging to the class 1
23 can be
obtained easily from (1.4.3) and (1.4.6) respectively just by putting the suitable values of the scalars 4 =
0,1,2,3 and 9. It is to be noted that any member of the class will not have MSE/minimum MSE smaller than
that of the proposed class of estimators 1
23.
It is to be noted that some empirical studies need to be conducted in order to find out situations where
some members of the proposed class 1
23 are found to be better than usual unbiased estimator , double sampling
ratio :3, product ;3, regression estimator ( „…3 = + †‡ # − # , †‡ being the sample estimate of the
population regression coefficient †), Srivastava (1970) estimator „3 and other existing estimators. Probably a
well designed Monte Carlo study will be quite useful and may throw some light on it.
Remark 1.4.1 In similar fashion the properties of the proposed class of estimator 1
23 can be discussed in the
case “when the second phase sample of size ! is drawn independently of the first phase sample of size ! ”.
References
Adhvaryu, D. and Gupta, P.C. (1983): On some alternative sampling strategies using auxiliary information.
Metrika, 30, 217-226.
Gangele, R. (1995): Study of some estimators using a priori/ auxiliary information. Ph. D. thesis submitted to
Vikram University, Ujjain, India.
Gupta, P.C. (1978): On some quadratic and higher degree ratio and product estimators. Jour. Ind. Soc. Agril.
Statist. , 30, 71-80.
Lui, K.J. (1990): Modified product estimators of finite population mean in finite sampling. Commun. Statist.
Theory Meth., 19(10), 3799-3807.
Prasad, B. (1989): Some improved ratio-type estimators of population mean and ratio in finite population sample
surveys. Commun. Statist. -Theory Meth. , 18,379-392.
Singh, H.P. (1986): Estimation of ratio, product and mean using auxiliary information in sample surveys. Alig.
Jour. Statist., 6, 32-44.
Singh, S. (2003): Advanced sampling theory with application. How Michael selected Amy. Vol. I and II, Kluwer
Academic Publishers, Dordrecht, The Netherlands.
Srivastava, S.K. (1970): Two-phase sampling estimator in sample surveys. Amt. Jour. Statist., 12, 23-27.
Srivastava, S.K. (1981). A general ized two-phase sampling estimator. Jour. Ind. Sac. Agri. Statist., 33, (1) , 38-
46.
Srivastava, V. K. (1974): On ratio method of estimation. Trab. Estadist. Invest. Operat. , 25, 113-117.
Upadhyaya, L.N., Singh, H.P. and Vos, J.W.E. (1985): On the estimation of population means and ratios using
supplementary information. Statistica Neerlandica, 39(3), 309-318.

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A note on estimation of population mean in sample survey using auxiliary information

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